What are the conditions for an optimal binary search tree and what is its advantage?

Question: What are the conditions for an optimal binary search tree and what is its advantage?

An optimal binary search tree (OBST), also known as a weighted binary search tree, is designed to minimize the expected search cost for a given set of keys and their access probabilities. The conditions for an OBST require a sorted list of keys and their corresponding probabilities or frequencies of access. The construction of an OBST is based on dynamic programming, where a table is built to contain the expected search costs for all possible subtrees derived from the original list of keys. The goal is to minimize the total cost, which is the sum of the products of each node's depth and its weight (probability). The advantage of an OBST is its efficiency in search operations, especially when the keys have varying probabilities of being searched. It ensures that the most frequently accessed items are closer to the root, thus reducing the average search time. This makes OBST particularly useful in applications like databases and compilers where certain items are accessed more frequently than others. The time complexity for constructing an OBST is generally O(n^3), but with optimizations, it can be reduced to O(n^2). Once constructed, the search operation in an OBST has a time complexity of O(log n), similar to a regular binary search tree, but with the added benefit of reduced average search time due to the optimized placement of nodes based on their access probabilities.

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